Having spun up both types of orgs just now, the difference is fairly immediately clear. The Enterprise Edition is your standard 30 day trial, and the Partner Enterprise Edition is a souped up special partner org.
Visually, indeed the PEE org follows the odd pattern of "Setup" being under your user drop down, and not it's own front link. Also, the setup menu is in the same order as PDEs, as opposed to normal DEs.
But the real kicker is in the capabilities.
The EE option gets you, as I said, a standard 30 day trial. It has 5 full Salesforce licenses, no Force.com sites, 5 Content licenses, 5 Apex Mobile licenses, and 5 Marketing user licenses.
The PEE option gets you a WHOLE LOT MORE.
It has 20 EE licenses, that expire in 3 years. It has another 20 Platform licenses, and all the various Portal, Community, and feature licenses you expect from a new partner dev org or org. It even has analytics cloud.
PEE also has 1 Dev Sandbox and 1 Full Sandbox. EE has NO sandboxes.
If you want to see what your app will look like in a brand new trial, go for EE. But if you want to do demos, DEFINITELY opt for PEE.